Problem: Simplify the following expression: $p = \dfrac{-120q^3 + 60q^2}{-108q^3 - 72q^2}$ You can assume $q \neq 0$.
Explanation: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $-120q^3 + 60q^2 = - (2\cdot2\cdot2\cdot3\cdot5 \cdot q \cdot q \cdot q) + (2\cdot2\cdot3\cdot5 \cdot q \cdot q)$ The denominator can be factored: $-108q^3 - 72q^2 = - (2\cdot2\cdot3\cdot3\cdot3 \cdot q \cdot q \cdot q) - (2\cdot2\cdot2\cdot3\cdot3 \cdot q \cdot q)$ The greatest common factor of all the terms is $12q^2$ Factoring out $12q^2$ gives us: $p = \dfrac{(12q^2)(-10q + 5)}{(12q^2)(-9q - 6)}$ Dividing both the numerator and denominator by $12q^2$ gives: $p = \dfrac{-10q + 5}{-9q - 6}$